How to get rid of white and black border on PNG
Blender's 3D view has some problems compositing planes with alpha in OpenGL mode. You can mitigate those effects to some extent (by changing the alpha clipping in the preferences or the different viewport alpha settings in material properties) but don't expect to have a perfect compositing of RGBA texture in the viewer, as Z compositing and Alpha compositing sometimes fight each other when selections are involved.

Render should turn out ok. If it doesn't, then there is a problem with your PNG images (wrong alpha association).

Wrong alpha association could be fixable (using the CM panel of the Image viewer if it) ONLY if the halos are the result of an incorrect alpha interpretation. If the image is encoded incorrectly, then there is no way to solve it, other than recreating the image textures properly.

Blender usually interprets PNG textures correctly if they are properly encoded, so check with a test render first. If it renders ok, and the halos pop up and disappear based on what's selected, then it's just the problem described first and you can't do really much about it other than trying to mitigate it as much as possible (which is probably a waste of time since it doesn't affect rendering)